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Li-Yau Inequality for Heat Equations on RCD∗(K,N) Metric Measure Spaces

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Abstract

In this paper, we will study the Li-Yau inequalities for weak solutions of the heat equation on RCD∗(K,N) metric measure spaces.

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References

  1. Ambrosio, L., Gigli, N., Savaré, G.: Bakry–Emery curvature-dimension condition and Riemannian Ricci curvature bounds. Ann. Probab. 43, 339–404 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L., Gigli, N., Savaré, G.: Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Invent. Math. 195(2), 289–391 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ambrosio, L., Mondino, A., Savaré, G.: On the bakry-Émery condition, the gradient estimates and the local-to global property of RCD∗(K,N) metric measure spaces. J. Geom. Anal. 26(1), 24–56 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bacher, K., Sturm, K.: Localization and tensonrization properties of the curvature-dimension for metric measure spaces. J. Funct. Anal. 259(1), 28–56 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bakry, D., Bolley, F., Gentil, I.: The Li-Yau inequality and applications under a curvature-dimension condition, available at arXiv:1412.5165

  6. Bakry, D., Ledoux, M.: A logarithmic Sobolev form of the Li-Yau parabolic inequality. Rev. Mat. Iberoam. 22(2), 683–702 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bauer, F., Horn, P., Lin, Y., Lippner, G., Mangoubi, D., Yau, S.-T.: Li-Yau inequality on graphs. J. Differ. Geom. 99, 359–405 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bakry, D., Qian, Z.: Harnack inequalities on a manifold with positive or negative Ricci curvature. Rev. Mat. Iberoam. 15(1), 143–179 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, Z., Yu, C., Zhao, F.: On the Li-Yau type gradient estimate of Li and Xu, arXiv:1706.06249 [math.DG]

  10. Davies, E.B.: Heat Kernels and Spectral Theory, Volume 92 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1989)

    Book  Google Scholar 

  11. Erbar, M., Kuwada, K., Sturm, K.: On the equivalence of the entropic curvature-dimension condition and bochners inequality on metric measure spaces. Invent. Math. 201, 993–1071 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Garofalo, N., Mondino, A.: Li-Yau and Harnack type inequalities in metric measure spaces. Nonlinear Anal. 95, 721–734 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gigli, N.: On the differential structure of metric measure spaces and applications. Mem. Amer. Math. Soc. 236(1113), 1–104 (2015)

  14. Hörmander, L.: The Analysis of Linear Partial Differential Operators I, 2th Edition, in ‘Grundlehren Der Mathematischen Wissenschaften’, vol. 256. Springer-Verlag, Berlin (1989)

    Google Scholar 

  15. Hamilton, R.: A matrix Harnack estimate for the heat equation. Commun. Anal. Geom. 1(1), 88–99 (1993)

    MathSciNet  Google Scholar 

  16. Jiang, R.: The Li-Yau inequality and heat kernels on metric measure spaces. J. Math. Pures Appl. 104(9), 29–57 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jiang, R., Zhang, H.C.: Hamiltons gradient estimates and a monotonicity formula for heat flows on metric measure spaces. Nonlinear Anal. 131, 32–47 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jiang, Y., Zhang, H.C.: Sharp spectral gaps on metric measure spaces. Calc. Var. PDE. 55(1, Art. 14), 14 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lee, P.W.Y.: Generalized Li-Yau estimates and Huisken’s monotonicity formula, available at arXiv:1211.5559

  20. Li, J., Xu, X.: Differential Harnack inequalities on Riemannian manifolds I: linear heat equation. Adv. Math. 226(5), 4456–4491 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, X.-D.: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pures Appl. 54, 1295–1361 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169, 903–991 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lott, J., Villani, C.: Weak curvature bounds and functional inequalities. J. Funct. Anal. 245(1), 311–333 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Olivé, X.R.: Neumann Li-Yau gradient estimate under integral Ricci curvature bounds, arXiv:1710.08649 [math.DG]

  25. Petrunin, A.: Alexandrov meets Lott–Villani–Sturm. Münster J. Math. 4, 53–64 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Qian, B.: Remarks on differential Harnack inequalities. J. Math. Anal. Appl. 409(1), 556–566 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Qian, Z., Zhang, H.-C., Zhu, X.-P.: Sharp spectral gap and Li-Yau’s estimate on Alexandrov spaces. Math. Z. 273(3-4), 1175–1195 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Souplet, P., Zhang, Q.S.: Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull. London Math. Soc. 38(6), 1045–1053 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sturm, K.: On the geometry of metric measure spaces. I, II. Acta Math. 196(1), 65–131, 133–177 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sturm, K.: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. 75, 273–297 (1996)

    MathSciNet  MATH  Google Scholar 

  31. Zhang, H.C., Zhu, X.P.: Ricci curvature on Alexandrov spaces and rigidity theorems. Comm. Anal. Geom. 18(3), 503–554 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang, H.C., Zhu, X.P.: Lipschitz continuity of harmonic maps between Alexandrov spaces, Invent. Math. https://doi.org/10.1007/s00222-017-0757-x (2017)

  33. Zhang, H.C., Zhu, X.P.: Local Li–Yau’s estimates on RCD∗(K,N) metric measure spaces. Calc. Var. PDE 55, 93 (2016). https://doi.org/10.1007/s00526-016-1040-5

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang, Q.S., Zhu, M.: Li-Yau gradient bound for collapsing manifolds under integral curvature condition. Proc. Amer. Math. Soc. 145(7), 3117–3126 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang, Q.S., Zhu, M.: Li-Yau gradient bounds under nearly optimal curvature conditions, arXiv:1511.00791 [math.DG]

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Acknowledgements

The author gratefully acknowledges support from the China Postdoctoral Science Foundation, grant number KLH1411048.

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  1. School of Mathematical Sciences, Fudan University, Shanghai, China

    Jia-Cheng Huang

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Huang, JC. Li-Yau Inequality for Heat Equations on RCD∗(K,N) Metric Measure Spaces. Potential Anal 53, 315–328 (2020). https://doi.org/10.1007/s11118-019-09770-2

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